Vladimir A. Boichenko, Gennadij A. Leonov, Volker Reitmann Dimension Theory for Ordinary Differential Equations Teubner
Contents Singular values, exterior calculus and Lozinskii-norms 15 1 Singular values and covering of ellipsoids 15 1.1 Introduction 15 1.2 Definition of singular values 17 1.3 Lemmas on covering of ellipsoids 19 2 Singular value inequalities 21 2.1 The Fischer-Courant theorem 21 2.2 The Binet-Cauchy theorem 24 2.3 The inequalities of Horn, Weyl and Fan 28 3 Compound matrices 30 3.1 Multiplicative compound matrices 30 3.2 Additive compound matrices 35 3.3 Applications to stability theory 38 4 Logarithmic matrix norms 40 4.1 Lozinskii's theorem 40 4.2 Generalization of the Liouville equation 45 5 The Yakubovich-Kalman frequency theorem 49 5.1 The frequency theorem for ODE's 49 5.2 The frequency theorem for discrete-time systems 52 6 Frequency-domain estimation of singular values 54 6.1 Linear differential equations 54 6.2 Linear difference equations 58 7 Exterior calculus in linear spaces 62 7.1 Multiplicative and additive compounds of operators... 62 7.2 Singular values of an operator acting between Euclidean spaces 72 7.3 Lemmas on covering of ellipsoids in an Euclidean space. 75 7.4 Singular value inequalities for operators 76
10 CONTENTS 11 Attractors, stability and Lyapunov functions 79 1 Dynamical systems, limit sets and attractors 79 1.1 Dynamical systems in metric spaces 79 1.2 Minimal global attractors 85 1.3 Time-invariant control systems 89 2 Dissipativity 91 2.1 Dissipativity in the sense of Levinson 91 2.2 Dissipativity and completeness of the Lorenz system... 92 2.3 Lyapunov-type results for dissipativity 96 2.4 Convergence in systems with several equilibrium states. 100 3 Stability of motion 106 3.1 Lyapunov stability 107 3.2 Orbital stability 115 3.3 Zhukovskii stability 119 4 Existence of a homoclinic orbit in the Lorenz system 124 4.1 Introduction 124 4.2 Estimates for the shape of global attractors 125 4.3 The existence of homoclinic orbits 127 5 The generalized Lorenz system 132 5.1 Definition of the system 132 5.2 Equilibrium states 133 5.3 Global asymptotic stability 134 5.4 Dissipativity 136 6 Orbital stability for flows on manifolds 138 6.1 Introduction 138 6.2 Dynamical systems with a local contraction property.. 139 6.3 The Andronov-Vitt theorem 143 6.4 Various types of variational equations 144 6.5 Asymptotic orbital stability conditions 147 6.6 Estimating the singular values and orbital stability... 162 6.7 Frequency-domain conditions for orbital stability in feedback control equations on the cylinder 169 III Introduction to dimension theory 175 1 Topological dimension 175 1.1 The inductive topological dimension 176 1.2 The covering dimension 182 2 Hausdorff and fractal dimensions 186 2.1 The Hausdorff measure and the Hausdorff dimension..186 2.2 Fractal dimension and lower box dimension 196
CONTENTS 11 2.3 Self-similar sets 202 2.4 Dimension of Cartesian products 205 3 Topological entropy 207 3.1 The Bowen-Dinaburg definition 208 3.2 The characterization by open covers 211 3.3 Some properties of the topological entropy 214 4 Dimension-like characteristics 219 4.1 Caratheodory measure, dimension and capacity 219 4.2 Properties of the Caratheodory dimension and Caratheodory capacity 223 IV Dimension and Lyapunov functions 229 1 Estimation of the topological dimension 229 1.1 Hilmy's theorem 229 1.2 Minimal sets for almost periodic flows 230 1.3 The frequency spectrum of almost periodic solutions.. 236 1.4 Frequency-domain conditions for upper topological dimension estimates of orbit closures 242 2 Upper estimates for the Hausdorff dimension 244 2.1 The limit theorem for Hausdorff measures 244 2.2 Corollaries of the limit theorem for Hausdorff measures. 250 2.3 Application of the limit theorem to the Henon map... 255 3 The application of the limit theorem to ODE's 260 3.1 An auxiliary result 260 3.2 Estimates of the Hausdorff measure and of Hausdorff dimension 262 3.3 The generalized Bendixson criterion 266 3.4 On the finiteness of the number of periodic solutions.. 267 3.5 Convergence theorems 268 4 Convergence in third-order nonlinear systems 269 4.1 The generalized Lorenz system 269 4.2 Euler's equations describing the rotation of a rigid body in a resisting medium 275 4.3 A nonlinear system arising from fluid convection in a rotating ellipsoid 276 4.4 A system describing the interaction of three waves in plasma 277 5 Estimates of fractal dimension 280 5.1 Maps with a constant Jacobian 280
12 CONTENTS 5.2 Autonomous differential equations which are conservative on the invariant set 283 6 Estimates of the topological entropy 285 6.1 Ito's generalized entropy estimate for maps 285 6.2 Application to differential equations 290 7 Fractal dimension estimates 292 7.1 The Rossler system 293 7.2 Lorenz equation 295 7.3 Equations of the third order 302 7.4 Equations describing the interaction between waves in plasma 307 8 Upper Lyapunov dimension 310 8.1 Definition of local Lyapunov exponents 310 8.2 An upper estimate for the upper Lyapunov dimension of the attractors of the Lorenz System 313 9 Formulas for the Lyapunov dimension 318 9.1 General results 318 9.2 The Henon map 325 9.3 Lorenz system 328 10 Invariant sets of vector fields 331 10.1 Introduction 331 10.2 Hausdorff dimension bounds for invariant sets of maps on manifolds 332 10.3 Time-dependent vector fields on manifolds 337 10.4 Convergence for autonomous vector fields 343 11 Use of a tubular Caratheodory structure 346 11.1 The system in normal variation 346 11.2 Tubular Caratheodory structure 350 11.3 Dimension estimates for sets which are negatively invariant for a flow 352 11.4 Flow invariant sets with an equivariant tangent bundle splitting 359 11.5 Generalizations of the theorems of Hartman-Olech and Borg 362 12 The Lyapunov dimension as upper bound 364 12.1 Statement of the results 364 12.2 Proof of Theorem 12.1.1 365 12.3 Global Lyapunov exponents and upper Lyapunov dimension 372 12.4 Application to the Lorenz system 373
CONTENTS 13 13 Lower estimates of the dimension of B-attractors 376 13.1 Introduction 376 13.2 Frequency-domain conditions for lower topological dimension bounds of global #-attractors 376 13.3 Lower estimates of the Hausdorff dimension of global B-attractors 381 13.4 Lower dimension estimates for global attractors based on the evolution of currents 382 A Some tools 385 A.I Definition of a differentiable manifold 385 A.2 Tangent space, tangent bundle and differential 387 A.3 Tensor products, exterior products and tensor fields... 389 A.4 Riemannian manifolds 390 A.5 Covariant derivative 392 A.6 Vector fields 392 A.7 Spaces of vector fields and maps 394 A.8 Parallel transport, geodesies and exponential map... 397 A.9 Curvature and torsion 398 A.10 Fiber bundles and distributions 400 A.ll Recurrence and hyperbolicity in dynamical systems... 402 A.12 Homology theory 403 A.13 Degree theory 405 A.14 Simple 5-linked parameterized m-boundaries 407 A.15 Geometric measure theory 409 A.16 Totally ordered sets 411 A.17 Almost periodic functions 412 Bibliography 415 Index 435